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In this paper, we derive and analyse rigorously a mathematical model of control strategies (screening, education, health care and immunization) of HCV in a community with inflow of infected immigrants. Both qualitative and quantitative analysis of the model is performed with respect to stability of the disease free and endemic equilibria. The results show that the disease free equilibrium is locally stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Using Lyapunov method, endemic equilibrium is globally stable under certain conditions. Numerical simulation of the model is implemented to investigate the sensitivity of certain key parameters on the HCV model in a community with inflow of infected immigrants. However, analysis shows that screening, education, health care and immunization have the effect of reducing the transmission of the disease in the community.

Hepatitis C is a blood borne liver disease, caused by the Hepatitis C Virus (HCV), first identified by [

A mathematical model is proposed and analysed to study the effect of screening, education, health care and immunization on the spread of HCV disease in the community. The model has five epidemiological classes: The susceptible

The interaction between the classes is being assumed as follows: Exposed individuals, acute infected and chronic infected immigrants enter into the population with the rates

viduals are infected with the HCV virus at a rate

tact rate of individuals with acute and chronic hepatitis C respectively. It is assumed that the rate of contact of susceptibles with chronic individuals is much less than that of acute infectives

Taking into account the above considerations, we then have the following schematic flow diagram (

From the above flow chart, and with

the model will be governed by the following system of equations:

with nonnegative initial conditions and

where

The model system of Equations (2) will be analysed qualitatively to get insight into its dynamical features which will give a better understanding of the effects of screening, education, health care and immunization on the transmission of HCV infection in the population with inflow of infected immigrants. The threshold which governs elimination or persistence of HCV will be determined and studied. We begin by finding the invariant region and show that all solutions of system (2) are positive

In this section, a region in which solutions of the model system (2) are uniformly bounded is the proper subset

Let

Using Birkhoff and Rota’s theorem [

where

Thus, as

Hence,

Furthermore, existence, uniqueness and continuation of results for system (2) hold in this region.

Lemma 1: The region

Lemma 2: Let the initial data be

Proof:

From the first equation of the model system (2), we have

The Integration factor is

Equations for

Thus

In the absence of the disease, which implies that

In this section, the threshold parameter that governs the spread of a disease which is called the effective reproduction number is determined. Mathematically, it is the spectral radius of the next generation matrix [

This definition is given for the models that represent spread of infection in a population. It is obtained by taking the largest (dominant) Eigen value, (spectral radius) of

where

Therefore,

and

The partial derivatives if (6) and (7) with respect to

and

In the absence of the disease and when

Now, taking the inverse of matrix (9) leads to

where

The spectral radius (dominant eigenvalue) of the matrix

Hence, the effective reproduction number of the model system (2) is given by

The effective reproduction number

Theorem 1: The disease free equilibrium of the model system (2) is locally asymptotically stable if

Theorem 1 implies that HCV can be eliminated from the community when

From Equation (12), for

In the absence of interventions (screening, education, health care and immunization) that is

Thus

Local stability of disease free equilibrium

The local stability analysis of the Jacobian matrix (13) of the system (2) can be done by the trace/determinant method. Where by matrix

and

where

Hence

That is equivalent to

since

Thus,

Endemic equilibrium point

where,

and

where,

The equation,

However it is important to note that

Theorem 2: The HCV model with screening, education, health care and immunization interventions have:

i) Precisely one unique endemic equilibrium if

ii) Precisely one unique endemic equilibrium if

iii) Precisely two endemic equilibrium if

iv) None otherwise.

Theorem 3: A unique endemic equilibrium point,

The global stability of the endemic equilibrium

Theorem 4: If

Proof: To establish the global stability of the endemic equilibrium

By direct calculating the derivative of

or

where,

Thus if

In determining how best to reduce human mortality and morbidity due to HCV, we calculate the sensitivity indices of the basic reproduction number,

Numerical values of sensitivity indices of

Definition 1: The normalised forward sensitivity index of a variable “

Having an explicit formula for

respect to

Other indices

are obtained following the same method and tabulated as follows:

From

Parameter Symbol | Sensitivity Index |
---|---|

−0.666666666 | |

0.6658711218 | |

−0.520211814 | |

0.3341288785 | |

−0.251532594 | |

−0.104042362 | |

−0.100613037 | |

−0.0734173793 | |

0.05209089976 | |

−0.0303517690 | |

0.02807805705 |

endemicity of the disease as they have negative indices.

The specific interpretation of each parameter shows that, the most sensitive parameter is the control based on education, health care and immunization

In this section, we illustrate the analytical results of the study by carrying out numerical simulations of the model system (2) using the following estimated parameter values:

Figures 2(a)-(d) show the proportion of HCV exposed, infective populations (acute, chronic) and proportion of HCV infectives all plotted against the proportion of susceptible population. This shows the dynamic beha-

viour of the endemic equilibrium of the model system (2) using the estimated parameter values above.

The phase portrait in Figures 2(a)-(d) shows that for any initial starting point or initial value, the solution curves tend to the endemic equilibrium point

In Figures 3(a)-(d), the variation of proportions of exposed, recovered, acute and chronic infective populations for different rates of education, health care and immunization

Figures 3(a)-(d), shows that the infected population decreases as the control strategies (education, health care and immunization),

Figures 4(a)-(d) shows the variation of proportions of exposed, acute and chronic infective populations and recovered population for different rates of screening.

From Figures 4(a)-(d) we vary the screened rate of infected immigrants, and it is seen that as the degree of screening increases, the infected population decreases. The results further show that increasing the screening rate, decreases the severity of the epidemic. Once again this confirms that, screening can reduce the inflow of infected immigrants into the community.

In this paper, a mathematical model of control strategies of HCV in a community with inflow of infected immigrants been established. Both qualitative and numerical analysis of the model was done. The model incorporates the assumption that infected immigrants enter in the community. It is shown that there exists a feasible region where the model is well posed in which a unique disease free equilibrium point exists. The disease free and endemic equilibrium points were obtained and their stabilities investigated. The model showed that the disease free equilibrium is locally stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Using Lyapunov method, endemic equilibrium is globally stable under certain conditions. A sensitivity analysis shows that the control based on education, health care and immunization